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Finding Fair and Efficient Allocations When Valuations Don't Add Up Finding Fair and Efficient Allocations When Valuations Don’t Add Up Nawal Benabbou, Mithun Chakraborty, Ayumi Igarashi, Yair Zick To cite this version: Nawal Benabbou, Mithun Chakraborty, Ayumi Igarashi, Yair Zick. Finding Fair and Efficient Allo- cations When Valuations Don’t Add Up. The 13th Symposium on Algorithmic Game Theory (SAGT 2020), Sep 2020, Augsburg (Virtual Conference), Germany. pp.32-46, 10.1007/978-3-030-57980-7_3. hal-02955635 HAL Id: hal-02955635 https://hal.sorbonne-universite.fr/hal-02955635 Submitted on 2 Oct 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Finding Fair and Efficient Allocations When Valuations Don't Add Up? Nawal Benabbou1, Mithun Chakraborty2, Ayumi Igarashi3, and Yair Zick4 1 Sorbonne Universit´e,CNRS, Laboratoire d'Informatique de Paris 6, LIP6 F-75005, Paris, France [email protected] 2 University of Michigan, Ann Arbor, USA [email protected] 3 National Institute of Informatics, Tokyo, Japan ayumi [email protected] 4 University of Massachusetts, Amherst, USA [email protected] Abstract. In this paper, we present new results on the fair and ef- ficient allocation of indivisible goods to agents whose preferences cor- respond to matroid rank functions. This is a versatile valuation class with several desirable properties (monotonicity, submodularity) which naturally models a number of real-world domains. We use these prop- erties to our advantage; first, we show that when agent valuations are matroid rank functions, a socially optimal (i.e. utilitarian social welfare- maximizing) allocation that achieves envy-freeness up to one item (EF1) exists and is computationally tractable. We also prove that the Nash welfare-maximizing and the leximin allocations both exhibit this fair- ness/efficiency combination, by showing that they can be achieved by minimizing any symmetric strictly convex function over utilitarian op- timal outcomes. Moreover, for a subclass of these valuation functions based on maximum (unweighted) bipartite matching, we show that a leximin allocation can be computed in polynomial time. Keywords: Fair Division · Envy-Freeness · Submodularity · Dichoto- mous preferences · Matroid rank functions · Optimal welfare 1 Introduction Suppose that we are interested in allocating seats in courses to prospective stu- dents. How should this be done? On the one hand, courses offer limited seats and have scheduling conflicts; on the other, students have preferences over the classes that they take, which must be accounted for. Course allocation can be thought of as a problem of allocating a set of indivisible goods (course slots) to agents (students). How should we divide goods among agents with subjective valuations? Can we find a \good" allocation in polynomial time? ? This research was funded by an MOE Grant (no. R-252-000-625-133) and a Singapore NRF Research Fellowship (no. R-252- 000-750-733). Most of this work was done when Chakraborty and Zick were employed at National University of Singapore (NUS), and Igarashi was a research visitor at NUS. 2 Benabbou et al. These questions have been the focus of intense study in the CS/Econ community in recent years; several justice criteria as well as methods for computing allo- cations that satisfy them have been investigated. Generally speaking, there are two types of justice criteria: efficiency and fairness. Efficiency criteria are chiefly concerned with maximizing some welfare criterion, e.g. Pareto optimality (PO). Fairness criteria require that agents do not perceive the resulting allocation as mistreating them; for example, one might want to ensure that no agent wants another agent's assigned bundle [18]. This criterion is known as envy-freeness (EF); however, envy-freeness is not always achievable with indivisibilities: con- sider, for example, two students competing for a single course slot. Any student receiving this slot would envy the other (in our stylized example, there is just the one course with the one seat). A simple solution ensuring envy-freeness would be to withhold the seat alto- gether, not assigning it to either student. This solution, however, violates most efficiency criteria. Indeed, as observed by Budish [12], envy-freeness is not always achievable, even with the weakest efficiency criterion of completeness requiring that each item is allocated to some agent. However, a less stringent fairness no- tion | envy-freeness up to one good (EF1) | can be attained. An allocation is EF1 if for any two agents i and j, there is some item in j's bundle whose removal results in i not envying j. EF1 complete allocations always exist, and in fact, can be found in polynomial time [26]. While trying to efficiently achieve individual criteria is challenging in itself, things get really interesting when trying to simultaneously achieve multiple jus- tice criteria. Caragiannis et al. [13] show that when agent valuations are additive | i.e. every agent i values its allocated bundle as the sum of values of individual items | there exist allocations that are both PO and EF1. Specifically, these are allocations that maximize the product of agents' utilities | also known as the max Nash welfare (MNW). Further work [6] shows that such allocations can be found in pseudo-polynomial time. While encouraging, these results are limited to agents with additive valuations. In particular, they do not apply to settings such as the course allocation problem described above (e.g. being assigned two courses with conflicting schedules will not result in additive gain), or other set- tings we describe later on. In fact, Caragiannis et al. [13] left it open whether their result extends to other natural classes of valuation functions, such as the class of submodular valutions.5 At present, little is known about other classes valuation functions; this is where our work comes in. 1.1 Our contributions We focus on monotone submodular valuations with binary (or dichotomous) marginal gains, which we refer to as matroid rank valuations. In this setting, the added benefit of receiving another item is binary and obeys the law of dimin- ishing marginal returns. This is equivalent to the class of valuations that can 5 There is an instance of two agents with monotone supermodular/subadditive valu- ations where no allocation is PO and EF1 [13]. Finding Fair and Efficient Allocations When Valuations Don't Add Up 3 be captured by matroid constraints; namely, each agent has a different matroid constraint over the items, and the value of a bundle is determined by the size of a maximum independent set included in the bundle. Matroids offer a highly versatile framework for describing a variety of domains [29]. This class of valuations naturally arises in many practical applications, be- yond the course allocation problem described above (where students are limited to either approving/disapproving a class). For example, suppose that a gov- ernment body wishes to fairly allocate public goods to individuals of different minority groups (say, in accordance with a diversity-promoting policy). This could apply to the assignment of kindergarten slots to children from different neighborhoods/socioeconomic classes6 or of flats in public housing estates to applicants of different ethnicities [9, 8]. A possible way of achieving group fair- ness in this setting is to model each minority group as an agent consisting of many individuals: each agent's valuation function is based on optimally match- ing items to its constituent individuals; envy naturally captures the notion that no group should believe that other groups were offered better bundles (this is the fairness notion studied by Benabbou et al. [8]). Such assignment/matching- based valuations (known as OXS valuations [25]) are non-additive in general, and constitute an important subclass of submodular valuations. Matroid rank functions correspond to submodular valuations with binary (i.e. f0; 1g) marginal gains. The binary marginal gains assumption is best understood in context of matching-based valuations | in this scenario, it simply means that individuals either approve or disapprove of items, and do not distinguish between items they approve (we call OXS functions with binary individual preferences (0; 1)-OXS valuations). This is a reasonable assumption in kindergarten slot allocation (all approved/available slots are identical), and is implicitly made in some public housing mechanisms (e.g. Singapore housing applicants are required to effec- tively approve a subset of flats by selecting a block, and are precluded from expressing a more refined preference model). In addition, imposing certain constraints on the underlying matching problem retains the submodularity of the agents' induced valuation functions: if there is a hard limit due to a budget or an exogenous quota (e.g. ethnicity-based quotas in Singapore public housing; socioeconomic status-based quotas in certain U.S. public school admission systems) on the number of items each group is able or allowed to receive, then agents' valuations are truncated matching-based valuations. Such valuation functions are not OXS, but are still matroid rank functions. Since agents still have binary/dichotomous preferences over items even with the quotas in place, our results apply to this broader class as well. Using the matroid framework, we obtain a variety of positive existential and algorithmic results on the compatibility of (approximate) envy-freeness with welfare-based allocation concepts. The following is a summary of our main results (see also Table 1): 6 see, e.g. https://www.ed.gov/diversity-opportunity. 4 Benabbou et al.
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